Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4}$ |
Frobenius angles: | $\pm0.0540867239847$, $\pm0.445913276015$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\zeta_{8})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $68$ | $626$ | $4624$ | $49282$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $10$ | $24$ | $54$ | $200$ | $730$ | $2240$ | $6494$ | $19392$ | $59050$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+x^3+x^2+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |
The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 2.9.a_ao and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.